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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrssre | Structured version Visualization version GIF version |
Description: A subset of extended reals that does not contain +∞ and -∞ is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
xrssre.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
xrssre.2 | ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) |
xrssre.3 | ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) |
Ref | Expression |
---|---|
xrssre | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrssre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
2 | ssxr 10699 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
4 | 3orass 1087 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) | |
5 | 3, 4 | sylib 221 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) |
6 | 5 | orcomd 868 | . 2 ⊢ (𝜑 → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ)) |
7 | xrssre.2 | . . . 4 ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) | |
8 | xrssre.3 | . . . 4 ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) | |
9 | 7, 8 | jca 515 | . . 3 ⊢ (𝜑 → (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) |
10 | ioran 981 | . . 3 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) | |
11 | 9, 10 | sylibr 237 | . 2 ⊢ (𝜑 → ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
12 | df-or 845 | . . 3 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) ↔ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) | |
13 | 12 | biimpi 219 | . 2 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) |
14 | 6, 11, 13 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 ∈ wcel 2111 ⊆ wss 3881 ℝcr 10525 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 |
This theorem is referenced by: supminfxr2 42108 |
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